Systems and methods herein generally relate to the problem of calculating powerflow studies of electrical networks, and more particularly to methods for incorporating and enforcing the physical limits of system controls in those studies.
Powerflow studies need to incorporate the effects of several kinds of automatic regulating devices that are always present in the operation of electrical networks. A non-exhaustive list of the most interesting ones, from the point of view of steady-state powerflows, comprises: voltage regulation by generators (AVR, Automatic Voltage Regulation) or by transformers (ULTC, Under-Load Tap Changers), frequency regulation by generators participating in AGC (Automatic Generation Control), real power regulation by phase shifters, reactive power regulation by transformers, and net real power regulation across tie lines (area interchange schedules). Additionally, controls may be local or remote. Some remote controls may be real, enabled by modern fast telecommunications; others are just convenient artifacts used in the context of planning studies (e.g. area interchange schedules). Whatever the case, a power flow method needs to incorporate all these types of control in order to be useful for real work. One crucial aspect, which is the subject of the present invention, is the correct enforcement of control limits. For instance, automatic voltage regulation in a generator is limited by the minimum and maximum values of reactive power output; when those limits are reached, the control resource “saturates” and can no longer sustain the desired voltage setpoint.
Traditional iterative powerflows deal with controls using either one of these two general approaches: (a) incorporating additional equations and new control variables into the definition of the matrix of the method (possibly eliminating some variables if they are given directly by the control setpoints); (b) keeping the original equations and using an “outer loop” approach, whereby the control variables are adjusted in between iterations, in proportion to the residuals of the regulated magnitude. The proportionality coefficients for these adjustments, the so-called sensitivities, are obtained either by theoretical modeling, direct computation, or empirical tests. This option (b) is favored by methods that keep the Jacobian constant through the iterations, such as the FDLF (Fast Decoupled Load Flow) method of Stott and Alsac, because it allows taking into account control limits in a fast manner. This, however, makes convergence behavior even harder to model and analyze. By contrast, the treatment of limits in method (a) requires a change in the equations in between iterations (for instance, a PV to PQ bus-type switch), so it would be more suited for a full NR (Newton Raphson) method.
Enforcing control limits raises several challenges in powerflow studies. The main one is that it makes the solution harder to arrive at. Iterative methods generally will take more iterations to converge, and moreover, subtle interactions between controls (due to the nonlinear nature of the problem) may conspire to produce “oscillations”, in which one or more controls have to bounce off their limits from one iteration to the next, therefore preventing convergence. Over the years, many heuristic techniques have been devised to improve the convergence rate of the industry-standard Newton-Raphson and Fast Decoupled powerflows in the presence of limits, with varying degrees of success. To date, it is a fair assessment to say that this problem has not been solved, as it keeps appearing in the everyday work of powerflow practitioners.
Additionally, most efforts have been devoted only to improving the chances of convergence, accelerating the convergence rate, and avoiding oscillations between saturated and non-saturated states. In contrast, very little has been done to address the problem of choosing among different configurations of saturated and non-saturated control states. In other words, due to nonlinearities, the problem may have multiple non-equivalent solutions, and therefore some sort of criterion is needed in order to select one. The trouble with both iterative approaches (a) and (b) mentioned above is that this issue is not even contemplated: the decisions as to which controls should reach saturation and which should not, are just the result of the unpredictable “dynamics” of the numerical iteration. The end result is that the inherent problems of fractality in iterative methods (which may cause divergence, or landing on a low-voltage solution) are then compounded by this somewhat arbitrary choice of saturated controls, dictated purely by the numerics.
Note that in cases in which there exists any additional information to help decide the relative priorities for saturation, such information may be trivially incorporated by powerflow methods. For instance, information about the timing response of controls; or about the operational details of their mutual coordination. But the invention disclosed here is concerned with the general scenario in steady-state powerflow studies, where none of this information is available and therefore this selection problem arises.
In contrast with iterative powerflow methods, U.S. Pat. Nos. 7,519,506 and 7,979,239 to Trias, take a very different approach to controls. The method, from here onwards termed the Holomorphic Embedding Load-flow Method (HELM), is non-iterative, constructive, and takes advantage of the specific mathematical structure of the power flow problem by using the techniques of Complex Analysis. Whereas iterative powerflows combine controls and their associated limits under the same treatment, HELM deals with automatic controls using a two-layered approach. The first layer incorporates control equations considering unlimited controlling resources, therefore using equality constraints for the setpoints. The smooth properties of these equality constraints allow them to be holomorphically embedded, thus preserving all the nice deterministic properties of the HELM core method. The second layer, which is the subject of the present invention, takes care of control limits by adding the inequality constraints for the control resources. This second problem is solved as an optimization problem, defining a suitable functional that is rooted in the physical insights gained by the underlying HELM method. As such, this optimization method is out of the core HELM methodology and therefore it may be applied to other powerflow methods in order to deal with control limits.
Seemingly related methods come from the area known as “Optimal Power Flow” (OPF). However, the problem of OPF is quite different from the standard steady-state powerflow one. Many of the magnitudes that are given as fixed parameters in a powerflow study become free variables in OPF, subject to global optimization under some prescribed functional and constraints. By contrast, a powerflow study had always been seen as a problem in solving a system of equations for the solution, implicitly assuming that there could only be one. The first innovative aspect of the invention herein disclosed is realizing the fact that the powerflow problem with control limits is really a problem in optimization. The second innovative aspect is the construction of the actual criterion to be used for the optimization.